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Chapter 12: Q. 12 (page 953)

Use your definition from Exercise 11 to show that the directional derivative of a function of a single variable f(x)at a point cin the direction of iis, f'(c)and that the directional derivative of fat cin the direction of -iis -f'(c).

Short Answer

Expert verified

It shows that, when u=-ithen the value isDuf(c)=-f'(c)

Step by step solution

01

Introduction

y=f(x)be a single variable function. The derivative of f(x)at a point cin limit form is given by

f'(c)=limh鈫�0f(c+h)-f(c)h

The direction derivative of fin the direction of the unit vector u=伪iat a point cis given by

Duf(c)=limh鈫�0f(c+伪h)-f(c)h

u=i, then 伪=1

Duf(c)=limh鈫�0f(c+h)-f(c)h

=f'(c)[From (1)]

u=i

Dnf(c)=f'(c)

02

Step 2:  Explanation

u=-iAgain when u=ithen 伪=-1-, so

Duf(c)=limh鈫�0f(c-h)-f(c)h

=limh鈫�0f(c+(-h))-f(c)h

畏=-h

Duf(c)=lim畏鈫�0f(c+畏)-f(c)(-畏)

=-lim畏鈫�0f(c+畏)-f(c)畏

=-f'(c)[From (1)]

u=-i

Duf(c)=-f'(c)

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